66
2 Size-Dependent Theories of Beams, Plates and Shells
196. Zhao, J., Zhou, S., Wang, B., Wang, X.: Nonlinear microbeam model based on strain gradient
theory. Appl. Math. Model. 36, 2674–2686 (2012)
197. Rajabi, F., Ramezani, S.: A nonlinear microbeam model based on strain gradient elasticity
theory. Acta Mech. Solida Sin. 26, 21–34 (2013)
198. Mohammadi, H., Mahzoon, M.: Thermal effects on postbuckling of nonlinear microbeams
based on the modified strain gradient theory. Compos. Struct. 106, 764–776 (2013)
199. Vatankhah, R., Kahrobaiyan, M.H., Alasty, A., Ahmadian, M.T.: Nonlinear forced vibration
of strain gradient microbeams. Appl. Math. Model. 37, 8363–8382 (2013)
200. Kahrobaiyan, M.H., Rahaeifard, M., Tajalli, S.A., Ahmadian, M.T.: A strain gradient functionally graded Euler-Bernoulli beam formulation. Int. J. Eng. Sci. 52, 65–76 (2012)
201. Akgoz, B., Civalek, O.: Buckling analysis of functionally graded microbeams based on the
strain gradient theory. Acta Mech. 224, 2185–2201 (2013)
202. Akgoz, B., Civalek, O.: Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM). Compos. B Eng. 55, 263–268 (2013)
203. Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T., Firoozbakhsh, K.: Strain gradient formulation of functionally graded nonlinear beams. Int. J. Eng. Sci. 65, 49–63 (2013)
204. Wang, B., Zhou, S., Zhao, J., Chen, X.: A size-dependent Kirchhoff micro-plate model based
on strain gradient elasticity theory. Eur. J. Mech. A Sol. 30, 517–524 (2011)
205. Ashoori, M.A., Mahmoodi, M.J.: A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory. Eur. J. Mech. A Sol. 40, 50–59 (2013)
206. Mohammadi, M., Fooladi, M.M.: An analytical solution for buckling analysis of sizedependent rectangular micro-plates according to the modified strain gradient and couple stress
theories. Acta Mech. 226, 3477–3493 (2015)
207. Mohammadi, M., Fooladi, M., Darijani, H.: Exact boundary conditions for buckling analysis
of rectangular micro-plates based on the modified strain gradient theory. Int. J. Multiscale
Comput. Eng. 13, 265–280 (2015)
208. Wang, B., Huang, S., Zhao, J., Zhou, S.: Reconsiderations on boundary conditions of Kirchhoff
micro-plate model based on a strain gradient elasticity theory. Appl. Math. Model. 40, 7303–
7317 (2016)
209. Zeighampour, H., Beni, Y.T.: Cylindrical thin-shell model based on modified strain gradient
theory. Int. J. Eng. Sci. 78, 27–47 (2014)
210. Allahbakhshi, A., Allahbakhshi, M.: Vibration analysis of nano-structure multilayered
graphene sheets using modified strain gradient theory. Front Mech. Eng. 10, 187–197 (2015)
211. Li, A., Zhou, S., Wang, B.: A size-dependent model for bi-layered Kirchhoff micro-plate
based on strain gradient elasticity theory. Compos. Struct. 113, 272–280 (2014)
212. Hosseini, M., Bahreman, M., Jamalpoor, A.: Using the modified strain gradient theory to
investigate the size-dependent biaxial buckling analysis of an orthotropic multi-microplate
system. Acta Mech. 227, 1621–1643 (2016)
213. Zhang, L., Liang, B., Zhou, S., Wang, B., Xue, Y.: An application of a size-dependent model on
microplate with elastic medium based on strain gradient elasticity theory. Mecc. 52, 251–262
(2016)
214. Chang, L., Rajapakse, R.K.N.D.: Continuum models incorporating surface energy for static
and dynamic response of nanoscale beams. IEEE Trans. Nanotechnol. 9(4), 422–431 (2010)
215. Gao, X.L., Mahmoud, F.F.: A new Bernoulli-Euler beam model incorporating microstructure
and surface energy effects. Z. Angew. Math. Phys. 65(2), 393–404 (2014)
216. Shaat, M., Mahmoud, F.F., Gao, X.L., Faheem, A.F.: Size dependent bending analysis of
Kirchhoff nano-plates based on a modified couple-stress theory including surface effects. Int.
J. Mech. Sci. 79, 31–37 (2014)
217. Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient
elasticity theory. Eur. J. Mech. A/Sol. 29, 591–599 (2010)
218. Zhang, L., Wang, B., Zhou, S., Xue, Y.: Modeling the size-dependent nanostructures: Incorporating the bulk and surface effects. J. Nanomech. Micromech. 7(2), 04016012 (2017)
219. Yan, Z., Jiang, L.: Influence of surface effects and flexoelectricity on vibration of piezoelectric
nanobeams. In: 13th International Conference on Fracture, Beijing, China (2013). Accesssed
June 16–21, 2013
2 Size-Dependent Theories of Beams, Plates and Shells
196. Zhao, J., Zhou, S., Wang, B., Wang, X.: Nonlinear microbeam model based on strain gradient
theory. Appl. Math. Model. 36, 2674–2686 (2012)
197. Rajabi, F., Ramezani, S.: A nonlinear microbeam model based on strain gradient elasticity
theory. Acta Mech. Solida Sin. 26, 21–34 (2013)
198. Mohammadi, H., Mahzoon, M.: Thermal effects on postbuckling of nonlinear microbeams
based on the modified strain gradient theory. Compos. Struct. 106, 764–776 (2013)
199. Vatankhah, R., Kahrobaiyan, M.H., Alasty, A., Ahmadian, M.T.: Nonlinear forced vibration
of strain gradient microbeams. Appl. Math. Model. 37, 8363–8382 (2013)
200. Kahrobaiyan, M.H., Rahaeifard, M., Tajalli, S.A., Ahmadian, M.T.: A strain gradient functionally graded Euler-Bernoulli beam formulation. Int. J. Eng. Sci. 52, 65–76 (2012)
201. Akgoz, B., Civalek, O.: Buckling analysis of functionally graded microbeams based on the
strain gradient theory. Acta Mech. 224, 2185–2201 (2013)
202. Akgoz, B., Civalek, O.: Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM). Compos. B Eng. 55, 263–268 (2013)
203. Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T., Firoozbakhsh, K.: Strain gradient formulation of functionally graded nonlinear beams. Int. J. Eng. Sci. 65, 49–63 (2013)
204. Wang, B., Zhou, S., Zhao, J., Chen, X.: A size-dependent Kirchhoff micro-plate model based
on strain gradient elasticity theory. Eur. J. Mech. A Sol. 30, 517–524 (2011)
205. Ashoori, M.A., Mahmoodi, M.J.: A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory. Eur. J. Mech. A Sol. 40, 50–59 (2013)
206. Mohammadi, M., Fooladi, M.M.: An analytical solution for buckling analysis of sizedependent rectangular micro-plates according to the modified strain gradient and couple stress
theories. Acta Mech. 226, 3477–3493 (2015)
207. Mohammadi, M., Fooladi, M., Darijani, H.: Exact boundary conditions for buckling analysis
of rectangular micro-plates based on the modified strain gradient theory. Int. J. Multiscale
Comput. Eng. 13, 265–280 (2015)
208. Wang, B., Huang, S., Zhao, J., Zhou, S.: Reconsiderations on boundary conditions of Kirchhoff
micro-plate model based on a strain gradient elasticity theory. Appl. Math. Model. 40, 7303–
7317 (2016)
209. Zeighampour, H., Beni, Y.T.: Cylindrical thin-shell model based on modified strain gradient
theory. Int. J. Eng. Sci. 78, 27–47 (2014)
210. Allahbakhshi, A., Allahbakhshi, M.: Vibration analysis of nano-structure multilayered
graphene sheets using modified strain gradient theory. Front Mech. Eng. 10, 187–197 (2015)
211. Li, A., Zhou, S., Wang, B.: A size-dependent model for bi-layered Kirchhoff micro-plate
based on strain gradient elasticity theory. Compos. Struct. 113, 272–280 (2014)
212. Hosseini, M., Bahreman, M., Jamalpoor, A.: Using the modified strain gradient theory to
investigate the size-dependent biaxial buckling analysis of an orthotropic multi-microplate
system. Acta Mech. 227, 1621–1643 (2016)
213. Zhang, L., Liang, B., Zhou, S., Wang, B., Xue, Y.: An application of a size-dependent model on
microplate with elastic medium based on strain gradient elasticity theory. Mecc. 52, 251–262
(2016)
214. Chang, L., Rajapakse, R.K.N.D.: Continuum models incorporating surface energy for static
and dynamic response of nanoscale beams. IEEE Trans. Nanotechnol. 9(4), 422–431 (2010)
215. Gao, X.L., Mahmoud, F.F.: A new Bernoulli-Euler beam model incorporating microstructure
and surface energy effects. Z. Angew. Math. Phys. 65(2), 393–404 (2014)
216. Shaat, M., Mahmoud, F.F., Gao, X.L., Faheem, A.F.: Size dependent bending analysis of
Kirchhoff nano-plates based on a modified couple-stress theory including surface effects. Int.
J. Mech. Sci. 79, 31–37 (2014)
217. Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient
elasticity theory. Eur. J. Mech. A/Sol. 29, 591–599 (2010)
218. Zhang, L., Wang, B., Zhou, S., Xue, Y.: Modeling the size-dependent nanostructures: Incorporating the bulk and surface effects. J. Nanomech. Micromech. 7(2), 04016012 (2017)
219. Yan, Z., Jiang, L.: Influence of surface effects and flexoelectricity on vibration of piezoelectric
nanobeams. In: 13th International Conference on Fracture, Beijing, China (2013). Accesssed
June 16–21, 2013